Math 6410 • ODE

\begin{enumerate}
\item[(a)] Show the uniqueness of Robin's problem
\[
  \begin{aligned}
    \Delta u&=f(x)\ [x\in D],\\
    \nabla u\cdot\mathbf{n}+a(x)u&=h(x)\ [x\in\partial D],\qquad
    [D\text{ is a 3D domain, }a(x)>0].
  \end{aligned}
\]
\item[(b)] When \(a\equiv0\), we have the Neumann problem. Show that it has no
solution, unless
\[
  \int_D f\dd x=\oint_{\partial D}h\dd A.
\]
Describe physical meaning of this fact in the case of steady heat conduction.
\item[(c)] Prove the uniqueness of the Neumann problem up to an additive constant.
\end{enumerate}